Abstract
The value function of a control problem does not satisfy, in general, the corresponding Bellman equation in the classical sense. In this chapter we show that under rather general conditions it is a unique viscosity solution of the equation, the concept introduced and developed by Crandall and Lions [21].
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The notion of a viscosity solution for first order Hamilton–Jacobi equation was introduced by M.G. Crandall and P.-L. Lions in [21] and extensively studied there. P.-L. Lions in [67] applied it to deterministic control problems. In our presentation we follow J. Yong and X.Y. Zhou [102] and W.H. Fleming and H.M. Soner [38]. For a connection between the theory of viscosity solution and the nonsmooth analysis see the book [17] by F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski.
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Zabczyk, J. (2020). Viscosity solutions of Bellman equations. In: Mathematical Control Theory. Systems & Control: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-44778-6_10
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DOI: https://doi.org/10.1007/978-3-030-44778-6_10
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-44776-2
Online ISBN: 978-3-030-44778-6
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